David Little
Mathematics Department
Penn State University
Eberly College of Science
University Park, PA 16802
Office: 403 McAllister
Phone: (814) 865-3329
Fax: (814) 865-3735
e-mail:dlittle@psu.edu

The Mean Value Theorem For Derivatives

The Mean Value Theorem states that if f(x) is continuous on [a,b] and differentiable on (a,b) then there exists a number c between a and b such that

The following applet can be used to approximate the values of c that satisfy the conclusion of the Mean Value Theorem. Simply enter the function f(x) and the values a, b and c. The applet automatically draws the secant line through the points (a,f(a)) and (b,f(b)), the line through the point (c,f(c)) parallel to the secant line, and the line tangent to f(x) at x=c.

Adjust the value of c by dragging the red dot along the curve. When you think you've found a value of c that satisfies the conclusion of the mean value theorem, press the "Locate" button in the lower left corner of the applet. The applet will adjust your value of c to a more accurate value that satisfies the mean value theorem. Note that the tangent line is now parallel to the secant line.

The values a, b and c can be changed by simply typing a new value, such as "1.2345", "pi/2", "sqrt(5)+cos(3)", etc. You may also change these values by using the up/down arrow keys or dragging the corresponding point left or right. To move the center of the graph, simply drag any point to a new location. To label the x-axis in radians (i.e. multiples of pi), click on the graph and press "control-r". To switch back, simply press "control-r" again.

Here is a list of functions that can be used with this applet.